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IN THIS ISSUE

### Research Papers

J. Appl. Mech. 2017;84(6):061001-061001-11. doi:10.1115/1.4036355.

Elastic spherical shells loaded under uniform pressure are subject to equal and opposite compressive probing forces at their poles to trigger and explore buckling. When the shells support external pressure, buckling is usually axisymmetric; the maximum probing force and the energy barrier the probe must overcome are determined. Applications of the probing forces under two different loading conditions, constant pressure or constant volume, are qualitatively different from one another and fully characterized. The effects of probe forces on both perfect shells and shells with axisymmetric dimple imperfections are studied. When the shells are subject to internal pressure, buckling occurs as a nonaxisymmetric bifurcation from the axisymmetric state in the shape of a mode with multiple circumferential waves concentrated in the vicinity of the probe. Exciting new experiments by others are briefly described.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(6):061002-061002-16. doi:10.1115/1.4036308.

In order to develop an active nonlinear beam model, the beam's kinematics is examined in this paper, by employing the kinematic assumption of a rigid cross section during deformation. As a mathematical tool, the moving frame method, developed by Cartan (1869–1951) on differentiable manifolds, is utilized by treating a beam as a frame bundle on a deforming centroidal curve. As a result, three new integrability conditions are obtained, which play critical roles in the derivation of beam equations of motion. These integrability conditions enable the derivation of beam models in Part II, starting from the three-dimensional Hamilton's principle and the d'Alembert's principle of virtual work. To illustrate the critical role played by the integrability conditions, the variation of kinetic energy is computed. Finally, the reconstruction scheme for rotation matrices for given angular velocity at each time is presented.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(6):061003-061003-17. doi:10.1115/1.4036317.

Utilizing the kinematics, presented in the Part I, an active large deformation beam model for slender, flexible, or soft robots is developed from the d'Alembert's principle of virtual work, which is derived for three-dimensional elastic solids from Hamilton's principle. This derivation is accomplished by refining the definition of the Cauchy stress tensor as a vector-valued 2-form to exploit advanced geometrical operations available for differential forms. From the three-dimensional principle of virtual work, both the beam principle of virtual work and beam equations of motion with consistent boundary conditions are derived, adopting the kinematic assumption of rigid cross sections of a deforming beam. In the derivation of the beam model, Élie Cartan's moving frame method is utilized. The resulting large deformation beam equations apply to both passive and active beams. The beam equations are validated with the previously reported results expressed in vector form. To transform passive beams to active beams, constitutive relations for internal actuation are presented in rate form. Then, the resulting three-dimensional beam models are reduced to an active planar beam model. To illustrate the deformation due to internal actuation, an active Timoshenko beam model is derived by linearizing the nonlinear planar equations. For an active, simply supported Timoshenko beam, the analytical solution is presented. Finally, a linear locomotion of a soft inchworm-inspired robot is simulated by implementing active $C1$ beam elements in a nonlinear finite element (FE) code.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(6):061004-061004-8. doi:10.1115/1.4036345.

The size effect of nanoporous materials is generally believed to be caused by the large ratio of surface area to volume, so that it is also called surface effect. Based on a recently developed elastic theory, in which the surface effect of nanomaterials is characterized by the surface energy density, combined with two micromechanical models of composite materials, the surface effect of nanoporous materials is investigated. Closed-form solutions of both the effective bulk modulus and the effective shear one of nanoporous materials are achieved, which are related to the surface energy density of corresponding bulk materials and the surface relaxation parameter of nanomaterials, rather than the surface elastic constants in previous theories. An important finding is that the enhancement of mechanical properties of nanoporous materials mainly results from the compressive strain induced by nanovoid's surface relaxation. With a fixed volume fraction of nanovoids, the smaller the void size, the harder the nanoporous material will be. The results in this paper should give some insights for the design of nanodevices with advanced porous materials or structures.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(6):061005-061005-6. doi:10.1115/1.4036394.

In this work, we carried out bulge test for quantifying the viscoelastic properties of poly (vinyl alcohol) (PVA) thin films with custom-developed apparatus. A viscoelastic bulge deformation (VBD) model based on the elasticity–viscoelasticity correspondence principle and spherical cap equation is established to describe the bulge deformation of polymeric thin films. The VBD model can be used to determine the time-dependent modulus by bulge test for polymeric films. Uniaxial compressive relaxation test and PRONY series fitting method are used to define the constitutive parameters of the VBD equations. We presented two types of VBD models in frequency domain under linear loading and step loading conditions. Through inverse Laplace transformation, the proposed VBD model can effectively predict the bulge deformation of PVA hydrogel thin film. Numerical simulations are also conducted to validate the VBD model under step loading conditions. This work provides a methodology to characterize the viscoelastic properties of polymeric films by bulge test.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(6):061006-061006-10. doi:10.1115/1.4036416.

For both polyimide membranes in aerospace and graphene membranes in nanoelectronics with surface accuracy requirements, wrinkles due to the extreme out-of-plane flexibility yield inverse influences on the properties and applications of membranes. In this study, on the basis of discrete topology optimization, we propose a prenecking strategy by adopting elliptical free edges to suppress the stretch-induced wrinkling. This prenecking strategy with the computer-aided-design (CAD)-ready format is versatile to eliminate wrinkles in stretched membranes with clamped ends and achieve wrinkle-free performances. The wrinkle-free capability of the prenecking strategy, capable of satisfying the shape accuracy requirements, indicates that by suffering insignificant area loss, concerning of wrinkling problems in membranes is no further required. As compared with the existing researches focusing on studying wrinkling behaviors, the prenecking strategy offers a promising solution to the stretch-induced wrinkling problem by eliminating wrinkles through design optimization.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(6):061007-061007-7. doi:10.1115/1.4036476.

Originated from the art of paper cutting and folding, kirigami and origami have shown promising applications in a broad range of scientific and engineering fields. Developments of kirigami-inspired inverse design methods that map target three-dimensional (3D) geometries into two-dimensional (2D) patterns of cuts and creases are desired to serve as guidelines for practical applications. In this paper, using programed kirigami tessellations, we propose two design methods to approximate the geometries of developable surfaces and nonzero Gauss curvature surfaces with rotational symmetry. In the first method, a periodic array of kirigami pattern with spatially varying geometric parameters is obtained, allowing formation of developable surfaces of desired curvature distribution and thickness, through controlled shrinkage and bending deformations. In the second method, another type of kirigami tessellations, in combination with Miura origami, is proposed to approximate nondevelopable surfaces with rotational symmetry. Both methods are validated by experiments of folding patterned thin copper films into desired 3D structures. The mechanical behaviors of the kirigami designs are investigated using analytical modeling and finite element simulations. The proposed methods extend the design space of mechanical metamaterials and are expected to be useful for kirigami-inspired applications.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(6):061008-061008-11. doi:10.1115/1.4036455.

The existing commercial programs for simulation of hydraulic fracturing (aka fracking, or frac) of gas (or oil) shale predict parallel vertical cracks to spread in vertical parallel planes, with no lateral branching. These cracks emanate from the perforation clusters on the horizontal wellbore casing, typically spaced 10 m apart or more. For such a large spacing, the rate of gas production observed at the wellhead can be explained only upon making the hypothesis that the large-scale (or regional) permeability of shale is (even at 3 km depth) about 10,000 times higher than the gas permeability of shale measured in the lab on drilled (nondried) shale cores under confining pressures corresponding to shale at the depth of about 3 km. This hypothesis has recently been rendered doubtful by a new three-phase medium theory that takes into account the body forces due to pressure gradients of pore water diffusing into the pores. This theory predicts the fracking to produce a dense system of branched vertical hydraulic cracks with the spacing of about 0.1 m. This value matches the crack spacing deduced from the gas production rate at wellhead based on the actual lab-measured permeability. It is calculated that, to boost the permeability 10,000 times, the width of the pre-existing open (unfilled) natural cracks or joints (whose ages are distributed from one to several hundred million years) would have to be about 2.8 μm (not counting possible calcite deposits in the cracks). But this width is improbably high because, over the geologic time span, the shale must exhibit significant primary and secondary creep or flow. It is shown that the creep must close all the cracks tightly (except for residual openings of the order of 10 nm) even if the cracks are propped open by surface asperities. The inevitability of secondary creep (or steady-state flow) is explained theoretically by activation of new creep sites at stress concentrations caused by prior creep deformation. The time of transition from primary to secondary creep is taken equal to the Maxwell time estimate from geology. The overall conclusion is that the 10,000-fold increase of large-scale permeability is most likely not pre-existing but frac-induced. Although this conclusion will make little difference for long-term forecasts, it would make a major difference for the understanding and control of the frac process.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2017;84(6):061009-061009-17. doi:10.1115/1.4036439.

Geomaterials such as sedimentary rocks often contain fissures and cracks. Such secondary porosity will result in the so-called mesoscopic flow in wave propagation. Its presence is increasingly believed to be responsible for the significant wave energy loss in the seismic frequency band. In the present research, the double-porosity dual-permeability model is employed to describe such phenomena. Based on the model, we derive both the three-dimensional (3D) and two-dimensional (2D) dynamic Green's functions for the infinite space. The existence of reciprocity relation is demonstrated, which is used to deduce some interesting relations among Green's functions. These relations can serve as a consistency check on the obtained results. The Somigliana-type integral equations, the basis for the boundary element method (BEM), are also established. The complete list of Green's functions appearing in the integral equations is provided, which enables numerical implementation. Furthermore, the asymptotic behavior of the obtained solutions is discussed.

Commentary by Dr. Valentin Fuster

### Technical Brief

J. Appl. Mech. 2017;84(6):064501-064501-4. doi:10.1115/1.4036257.

A recently developed transfer printing technique, laser-driven noncontact microtransfer printing, which involves laser-induced heating to initiate the separation at the interface between the elastomeric stamp (e.g., polydimethylsiloxane (PDMS)) and hard micro/nanomaterials (e.g., Si chip), is valuable to develop advanced engineering systems such as stretchable and curvilinear electronics. The previous thermomechanical model has identified the delamination mechanism successfully. However, that model is not valid for small-size Si chip because the size effect is ignored for simplification in the derivation of the crack tip energy release rate. This paper establishes an accurate interfacial fracture mechanics model accounting for the size effect of the Si chip. The analytical predictions agree well with finite element analysis. This accurate model may serve as the theoretical basis for system optimization, especially for determining the optimal condition in the laser-driven noncontact microtransfer printing.

Commentary by Dr. Valentin Fuster